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Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

# Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

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Solving Inequalities, Compound Inequalities and Absolute Value Inequalities

Sec 1.5 &1.6

pg. 33 - 43

Objectives

The learner will be able to (TLWBAT): solve inequalities solve real-world problems with

inequalities solve compound inequalities solve absolute value inequalities

Properties

Trichotomy Property For any two real numbers a and b,

a < b a = b a > b

Properties of Inequalities

For any real number a, b, and c Addition Prop.

If a>b, then a + c > b + c If a<b, then a + c < b + c

Subtraction Prop. If a>b, then a – c > b – c If a<b, then a – c < b – c

See pg. 33

Work this problem

x – 7 > 2x + 2 -x -x -7 > x + 2 -2 -2 -9 > x Now we need to graph our answer on

a number line.

Graphing on a Number Line

Let’s look at our previous answer, -9 > x

-9 0

Are my “x’s” here?Or here?

-9 is GREATER than x, so my “x’s” that make senseare -10, -10.1, -11, etc. Anything smaller than -9

For < or > you use an opening dot or point -

For < or > you use a closed dot or point -

Properties of Inequalities

Multiplication Prop. For any real numbers

a, b, and c If c is positive If a>b, then ac>bc If a<b, then ac<bc

If c is negative If a>b, then ac<bc If a<b, then ac>bc

Division Prop. For any real numbers

a, b, and c If c is positive If a>b, then a/c>b/c

If a<b, then a/c<b/c

If c is negative If a>b, then a/c<b/c

If a<b, then a/c>b/c

Work this problem

3x – 7 < 7x + 13

-4x – 7 < 13

7 7

-4x < 20

-4 -4

x > -5

Now let’s graph

-7x -7x

0-5

{x | x > -5}

This is read as the set of all x such that x is greater than or equal to -5

Work this problem

3(a +4) – 2(3a +4) ≤ 4a - 1

3a + 12 – 6a – 8 ≤ 4a – 1

-3a + 4 ≤ 4a – 1

4 ≤ 7a - 1

5 ≤ 7a5/7 ≤ a

Now graph andexpress in setbuilder notation.

0

{a | a ≥ 5/7}

You can also use interval notation. Interval notation uses ( & ) for < or > and [ & ] for ≤ & ≥. We also use -∞ (negative infinity) & +∞ (positive infinity)

Interval notation for this problem would be [5/7, +∞)

Compound Inequalities

A compound inequality consists of two inequalities joined by the word “and” or the word “or”.

You must solve both inequalities and then graph. The final graph of the “and” inequality is the

intersection of both individual solution graphs. The final graph of the “or” inequality in the union

of both individual solution graphs

Let’s solve an “and” compound inequality

7 < 2x - 1 < 15

Method 1 – Divide into two problems

7 < 2x – 18 < 2x4 < x

Method 2 – Work together

2x – 1 < 152x < 16x < 8

7 < 2x – 1 < 158 < 2x < 164 < x < 8

Whateveryou do toone sidedo to the other!

Now Graph

4 < x

x < 8

0

0

0

4

4

4

8

8

8

{ x | 4 ≤ x < 8}

“Means 7 < 2x -1 and 2x – 1< 15”

4 < x < 8

Let’s work about the same problem as an “or” inequality

7 < 2x -1 or 2x – 1 < 15

We know the answer is 4 < x and x < 8, but thistime the answer graph is different. It is the UNIONof the two graphs.

4 < x

x < 8

0

0

0

4

4

8

8

The solution set is all real numbers .

4 < x or x < 8

Work this problem

2x + 7 < -1 or 3x + 7 > 10

2x + 7 < -1 3x + 7 > 10

2x < -8x < -4

3x > 3x > 1

0-4

0

or

1

0-4 1

Absolute Value Inequalities

|a| < b, where b > 0 work as an “and” problem -b < a < b

|a| > b, where b > 0 work as an “or” problem a > b or a < -b

Work these problems

|x – 1| < 3 -3 < x -1 < 3 -2 < x < 4

|x -1 | > 3 x -1 > 3 or x -1 < -3 x > 4 or x < -2

0 0

Absolute Value Inequalities

|a| < b, where b < 0 if b is less than zero, it is negative, so

there is no solution |a| > b, where b < 0

if b is less than zero, it is negative, so every real number is a solution or all reals.

Work these problems

|2x – 7| < -5

There is no solutionfor the above sincethe absolute value cannot be less than zero.

|3x – 1| + 9 > 2

|3x – 1| > -7

Any value of “x” willmake this statementtrue, since the absolute value is alwaysgreater than a negativenumber

All Real numbers